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A model to predict the plasma properties inside a thermionic hollow cathode as a function of operational conditions and geometry is presented. The hollow cathode features a lanthanum hexaboride (LaB6) insert, which is capable of emitting current densities as high as 10^5~\mathrmAm^-2 at temperatures of $\sim 1900$ K, along with a tantalum orifice plate located at the downstream end of the cathode tube. The model self-consistently computes the plasma parameters in both the emitter and orifice regions. A simple semiempirical relation is suggested to evaluate the plasma penetration depth in the cathode interior, which is of primary importance to establish the plasma conditions. The heat transfer mechanisms and the related temperature gradients along the cathode are evaluated with the aid of a dedicated thermal model, which is coupled to the plasma model and accounts for temperature-dependent material properties. A parametric study of the cathode performance was conducted to assess the dependence of the power consumption and operational lifetime on discharge current and mass flow rate, as well as on the geometry. The results are in good agreement with both theoretical and experimental trends found in the literature as well as with experimental data collected by Alta. Further developments will include a deeper investigation into the cathode erosion phenomena, along with a broader comparison with empirical data. [ABSTRACT FROM PUBLISHER]